On 1/19/2014 3:41 PM, Jason Resch wrote:
On Jan 19, 2014, at 3:31 PM, meekerdb <[email protected]> wrote:
On 1/19/2014 9:45 AM, Bruno Marchal wrote:
But why should that imply *existence*.
It does not. Unless we believe in the axioms, which is the case for elementary
arithmetic.
But what does "believe in the axioms" mean. Do we really believe we can *always* add
one more? I find it doubtful. It's just a good model for most countable things. So I
can believe the axioms imply the theorems and that "17 is prime" is a theorem, but I
don't think that commits me to any existence in the normal sense of "THAT exists".
Axioms are a human invention which only approach the truth that was already there. Our
picking some axioms to believe in changes nothing.
You seem not to appreciate that this dissipates the one essential advantage of
mathematical monism: we understand mathematics (because, I say, we invent it). But if
it's a mere human invention trying to model the Platonic ding and sich then PA may not be
the real arithmetic. And there will have to be some magic math stuff that makes the real
arithmetic really real.
Brent
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